C3 chapter 4 - numerical methods

In this chapter you will learn how to solve complicated equations by using numerical methods rather than purely algebraic methods.

Numerical methods is a phrase used to describe solving equations by using number approximations rather than purely algebraic methods. Mathematicians generally prefer to solve problems algebraically to get exact solutions, but some problems are difficult, or even impossible to solve exactly using algebra. For these problems (and in engineering and science contexts where an approximate answer is often good enough) a numerical method may be used.
You will have encountered a numerical method for solving equations at GCSE: 'trial and improvement.' At GCSE mathematics, an equation (usually a cubic equation or a simple equation involving reciprocal functions) is given and a solution is required to a given number of decimal places or significant figures. Here's a typical example:
The desired solution is an organised search for a value which is the closest value (to the required degree of precision, e.g. 1 decimal place) to the exact solution. Marks would often be missed or denied if the method is omitted.
Often a table is a good approach to show the steps:
Now this particular cubic equation can be solved algebraically, but you really wouldn't want to do that. You can see why if you look at the solution in the Word document below:

The solution of equations by numerical methods is deeply connected to the development of computing as this article from the University of Cambridge Researchshows: "This 'Mathematical Laboratory' began life as a two-man team, confined to the Anatomy School’s North Wing, and was charged with providing a resource for solving complex problems by 'numerical methods'."

C3 section 4.1

Prior knowledge required:
You will learn:

Examples:

C3 Exercise 4A

• C3 Exercise 4A worked solutions:
• Questions similar to this exercise: